Protesters hit Yemeni streets, despite concession from president

NEW: Saleh condemns the "wrong mobilization ... that resulted in social unrest"

NEW: The president hopes "political forces" respond to his not seeking re-election

Thousands rallied against the government, and pro-Saleh supporters held a counter-rally

The protests in Yemen come amid a string of uprisings in the region

Sanaa,Yemen (CNN) -- What seemed like thousands of anti-government protesters gathered near Sanaa University in Yemen's capital early Thursday morning, a clear indication that many in the country were not satisfied with President Ali Abdullah Saleh's recent announcement that he would not seek re-election.

Protesters of all ages chanted and held signs with messages against poverty and the government. Some proclaimed that Saleh needed to step down.

As the protest quickly grew, there was very little visible security in the area.

About a kilometer away from the anti-government protest, a large crowd of government supporters gathered for a counter demonstration.

Many in that crowd expressed support for Saleh and said he was doing a good job as president.

There were no apparent clashes between the two sides or with security forces.

Later in the day, in an apparent critique of his detractors, Saleh told members of his defense council that he wanted to "thank the Yemeni people who stayed calm despite wrong mobilization over the last months that resulted in social unrest in the Yemeni street," according to a report from the state-run SABA news service.

He called out the anti-government demonstrators for going ahead with their march Thursday despite his concession a day earlier.

"None reacted positively and parties continued to rallies and protests at a time when the public is nervous," Saleh said, according to SABA. The president added that he watched the demonstrations Thursday on TV.

Trying to quell a growing discontent in the country, Saleh said he will not seek re-election once his current term ends in 2013, after more than three decades in office. Saleh has been in office for 32 years and was last re-elected in 2006.

He won't install his son to replace him, he said. He also has asked his political opponents "to re-engage in dialogue in hopes of reaching a sustainable and reconcilable political agreement," the Yemeni government said.

"I have a big hope the political forces will respond to the initiative and take it seriously," Saleh said Thursday. "All should show interest in dismissing tensions and agitation because dialogue is the best way to address all issues."

Thursday's protest came amidst similar ongoing unrest in Egypt and a revolt in Tunisia that forced that nation's longtime strongman to flee to Saudi Arabia in mid-January.

King Abdullah of Jordan, meanwhile, has sacked his government and appointed a new prime minister in the face of protests there.

The protests -- which have also caught on to various extents in Algeria and Sudan -- have proved to be "a real watershed event for the Arab world," said Blake Hounshell, managing editor of Foreign Policy magazine. "It's really unprecedented."